The Sharpe ratio, written out
Sharpe takes the average return of a series, subtracts a risk free rate to get the excess, and divides by the standard deviation of those same returns. The numerator is reward. The denominator is total variation, measured without regard to direction. That last detail is the whole character of the measure. A month that beat the average by a wide margin raises the standard deviation exactly as much as a month that missed by the same margin, so a system producing occasional very large gains is penalised for producing them. Whether that is sensible depends on what you are measuring. For a portfolio where the mandate is steady compounding, punishing all variation is defensible. For a trading system whose edge is a small number of large winners paid for by many small losses, it describes the engine as a fault. Nothing is wrong with the formula. It simply answers a specific question, and that question is not always yours.
The Sortino ratio and the target you choose
Sortino keeps the same numerator and replaces the denominator with downside deviation. You pick a target, often zero or a minimum acceptable return, and you only square the shortfalls below it. Returns above the target contribute nothing to the risk measure, which matches the intuition that upside is not risk. There is a convention here that trips people up and changes the number materially. The standard form divides the sum of squared shortfalls by the total count of periods, not by the count of negative periods. If you divide by the negative count instead you will get a larger denominator and a smaller ratio, and two people quoting Sortino for the same series can then disagree without either being careless. Always state the target and the divisor. A Sortino figure without those two pieces of information is not comparable with anything.
A worked example, series A
Define twelve monthly returns yourself so there is nothing to argue about. Series A has six months at plus 3 per cent and six at minus 1 per cent. The sum is 18 minus 6, which is 12, so the mean is 1.0 per cent per month. Every deviation from that mean is either plus 2 or minus 2, so every squared deviation is 4, and the average squared deviation is also 4. The standard deviation is therefore 2.0. With a risk free rate of zero, Sharpe is 1.0 divided by 2.0, which is 0.50. Now the downside side. With a target of zero, the only shortfalls are the six months at minus 1, each contributing 1 when squared. The sum is 6, divided by all twelve periods gives 0.5, and the square root of 0.5 is 0.7071. Sortino is 1.0 divided by 0.7071, which is 1.414. Note the divisor: the variance used all twelve periods, and so did the downside calculation. Both ratios here are monthly figures rather than annual ones, which matters the moment you compare them against a published number.
The same arithmetic on series D, and the reversal
Series D also has twelve months and the same mean, built differently: ten months at plus 1.6 per cent and two at minus 2.0 per cent. The sum is 16 minus 4, which is 12, so the mean is again 1.0. Deviations are plus 0.6 ten times and minus 3.0 twice. Squared, that is 0.36 ten times, giving 3.6, plus 9 twice, giving 18, for a total of 21.6. Divide by twelve to get 1.8, and the standard deviation is 1.3416. Sharpe is 1.0 divided by 1.3416, which is 0.745. For the downside, only the two months at minus 2.0 count, each contributing 4, so the sum is 8, divided by twelve gives 0.6667, and the square root is 0.8165. Sortino is 1.0 divided by 0.8165, which is 1.225. Line the four figures up and the ranking flips. Series D wins on Sharpe, 0.745 against 0.50. Series A wins on Sortino, 1.414 against 1.225. Nothing was manipulated to get that reversal. Series D is steadier overall, since its monthly results sit closer together, and total deviation is all that Sharpe measures. When it does lose, though, it loses twice as deeply as series A ever does, and depth below zero is all that Sortino measures. Series A is the choppier series with the shallower losses. Choose the ratio first and you have chosen the winner.
Annualising, and what the square root assumes
Monthly ratios are usually quoted on an annual basis by multiplying by the square root of the number of periods in a year. For monthly data that is the square root of 12, roughly 3.4641. Series A becomes a Sharpe of about 1.73 and a Sortino of about 4.90. Series D becomes about 2.58 and about 4.24. The ranking is unchanged, because scaling both sides of a comparison by the same constant cannot change an ordering. What that scaling does assume is that returns are independent from period to period. If there is serial correlation, if good months tend to follow good months, the annualised figure is inflated and the inflation is invisible in the final number. Trading returns often do show streaks, since a market condition that suits a system persists for a while. Treat any annualised ratio built from a short series as an estimate with a wide error, and treat a very high one as a prompt to check the period count rather than celebrate.
The conventions that quietly change the number
Several defensible choices move these ratios, and an unstated choice makes two figures incomparable. The divisor for variance can be the number of periods or that number minus one, which matters most when the series is short. The risk free rate can be subtracted or ignored, and ignoring it raises Sharpe. The target for Sortino can be zero, the risk free rate, or a required return, and raising the target lowers the ratio. The period length changes everything, because daily data produces a different standard deviation from monthly data on the same equity curve, and annualising does not fully reconcile them. The return definition matters too, since percentage returns on a changing account balance behave differently from results measured in R. None of this is a reason to avoid the ratios. It is a reason to publish the recipe alongside the number, which is also good practice in a weekly review.
What neither ratio can see
Both measures compress a whole history into one figure, and several things that matter do not survive the compression. Neither sees order, so a series with all its losses clustered at the start scores the same as one with them spread out, while the two experiences are nothing alike. Neither reports drawdown directly, which is the number that actually decides whether a plan gets abandoned. Neither handles a fat tail well, because a single catastrophic period in a short series distorts both the mean and the deviation at once, and because the worst loss you have not met yet contributes nothing. Both are also unstable on small samples, which is the honest caveat: a ratio computed from twelve observations carries an error wide enough to swallow the difference between the two series above. Read them beside a drawdown figure and a trade count, keep your risk rules independent of them, and use something like a stretch measure when the question is about a single move rather than a record. Keeping forward results on the live chart is what eventually makes any of these figures mean something.
FAQ
Which ratio should I prefer for a trading system?
Report both, since they answer different questions. Sortino usually suits a system whose profits arrive in a few large winners, because Sharpe penalises those winners for being large. Sharpe suits a mandate where steadiness itself is the goal. The pair together is more informative than an argument about which single one is correct.
Can Sortino be lower than Sharpe?
Yes. If the target is set high enough, most periods fall short of it and the downside deviation can exceed the full standard deviation, which pushes Sortino below Sharpe. It also happens when losses are rare but very large relative to the spread of the gains. The target choice drives most of this.
How many periods do I need before the ratio means anything?
More than most people use. Twelve monthly observations give an estimate with an error wide enough to reverse a ranking, as the two worked series show. Treat any ratio from under three years of monthly data, or a few hundred trades, as a rough indication rather than a measurement you can defend.
Does a high Sharpe ratio mean low risk of a big loss?
No. It means variation around the average was small relative to that average over the measured period. A system can post a high figure for a long time by selling an unlikely event and then lose far more than the ratio ever suggested. The measure cannot see a loss that has not happened.
Should I use returns or R multiples in the calculation?
Either, as long as you say which and stay consistent. R multiples strip out position sizing and describe the system. Percentage returns include your sizing decisions and describe the account. The two can rank the same set of trades differently, so mixing them inside one comparison produces a meaningless number.
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